2 Answers
2

First of all, consider the grammatical class they belong to in order to understand usage better:

Parallel is an adjective (see below, if I edit it further, I need to change it too much), a noun (see for example each of the imaginary parallel circles of constant latitude on the earth's surface) or even a verb (be side by side with something extending in a line).

When it's an adjective, as you may already know, it's used to classify objects that put side by side have the same distance continuously everywhere.

But it has also other meanings: about Computing, for example, it means "involving the simultaneous performance of operations". It can be also related to Music or Grammar.

Parallesism is a noun and its adjective form is Parallelistic.

As F'x said, it means: "the state of being parallelorof corresponding in some way." Also in this case it can refer to other fields, such as computing or poetry.

Parallelism is the state of being parallel, or (in grammar or literature) the use of successive constructions with the same grammatical structure, or (in computer science) the use of parallel processing.

So, if you're discussing mathematics, computer science or literature, odds are you want parallelism. For everything else, parallel is more likely what you need.